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Properties

Label 96.17.h.a
Level $96$
Weight $17$
Character orbit 96.h
Self dual yes
Analytic conductor $155.832$
Analytic rank $0$
Dimension $1$
CM discriminant -24
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [96,17,Mod(17,96)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("96.17"); S:= CuspForms(chi, 17); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(96, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 17, names="a")
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 96.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-6561] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(155.831562102\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 6561 q^{3} - 31294 q^{5} - 11491202 q^{7} + 43046721 q^{9} + 345579838 q^{11} + 205319934 q^{15} + 75393776322 q^{21} - 151608576189 q^{25} - 282429536481 q^{27} - 444815014078 q^{29} - 913947924482 q^{31}+ \cdots + 14\!\cdots\!98 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0
0 −6561.00 0 −31294.0 0 −1.14912e7 0 4.30467e7 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.17.h.a 1
3.b odd 2 1 96.17.h.b 1
4.b odd 2 1 24.17.h.b yes 1
8.b even 2 1 96.17.h.b 1
8.d odd 2 1 24.17.h.a ✓ 1
12.b even 2 1 24.17.h.a ✓ 1
24.f even 2 1 24.17.h.b yes 1
24.h odd 2 1 CM 96.17.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.17.h.a ✓ 1 8.d odd 2 1
24.17.h.a ✓ 1 12.b even 2 1
24.17.h.b yes 1 4.b odd 2 1
24.17.h.b yes 1 24.f even 2 1
96.17.h.a 1 1.a even 1 1 trivial
96.17.h.a 1 24.h odd 2 1 CM
96.17.h.b 1 3.b odd 2 1
96.17.h.b 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 31294 \) acting on \(S_{17}^{\mathrm{new}}(96, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 6561 \) Copy content Toggle raw display
$5$ \( T + 31294 \) Copy content Toggle raw display
$7$ \( T + 11491202 \) Copy content Toggle raw display
$11$ \( T - 345579838 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 444815014078 \) Copy content Toggle raw display
$31$ \( T + 913947924482 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 95084003330878 \) Copy content Toggle raw display
$59$ \( T + 228567745583042 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 1517731242771838 \) Copy content Toggle raw display
$79$ \( T - 3006783469044478 \) Copy content Toggle raw display
$83$ \( T + 1495615549999682 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 886127550036478 \) Copy content Toggle raw display
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